One of the most exciting developments in string theory in the last years deals with dualities. In the area of string compactifications mirror symmetry plays an important role. There, string theories on so-called Calabi-Yau (CY) manifolds, which form a mirror pair, lead to the same physics. Such CY mirror pairs can be constructed in toric geometry as complete intersections (CICYs), whose information is encoded in pairs of reflexive polytopes. I wrote a C-program, with which one can search reflexive polyhedra for CICYs and compute their cohomology. Since nobody has examined toric CICYs in a systematically way, I present such examples and discuss the substantial differences to the case of hypersurfaces. Furthermore I use the mirror map to calculate explicitly the worldsheet instantons in one of those cases.